Cubic lattice sums include the following:
(1) | |||
(2) | |||
(3) |
(4) | |||
(5) | |||
(6) |
For hexagonal sums, Borwein and Borwein (1987, p. 292) give
(7) |
(8) |
See also Benson's Formula, Madelung Constants
References
Borwein, D. and Borwein, J. M. ``On Some Trigonometric and Exponential Lattice Sums.'' J. Math. Anal. 188, 209-218, 1994.
Borwein, D.; Borwein, J. M.; and Shail, R. ``Analysis of Certain Lattice Sums.'' J. Math. Anal. 143, 126-137, 1989.
Borwein, D. and Borwein, J. M. ``A Note on Alternating Series in Several Dimensions.'' Amer. Math. Monthly 93, 531-539, 1986.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mdlung/mdlung.html
Glasser, M. L. and Zucker, I. J. ``Lattice Sums.'' In Perspectives in Theoretical Chemistry: Advances and Perspectives 5,
67-139, 1980.
© 1996-9 Eric W. Weisstein